$\ell^\infty$-cohomology: amenability, relative hyperbolicity, isoperimetric inequalities and undecidability

Abstract

We revisit Gersten's $\ell^\infty$-cohomology of groups and spaces, removing the finiteness assumptions required by the original definition while retaining its geometric nature. Mirroring the corresponding results in bounded cohomology, we provide a characterization of amenable groups using $\ell^\infty$-cohomology, and generalize Mineyev's characterization of hyperbolic groups via $\ell^\infty$-cohomology to the relative setting. We then describe how $\ell^\infty$-cohomology is related to isoperimetric inequalities. We also consider some algorithmic problems concerning $\ell^\infty$-cohomology and show that they are undecidable. In an appendix, we prove a version of the de Rham's theorem in the context of $\ell^\infty$-cohomology.